
For electromagnetic modes with \(\ell = 2\)  and \(r_+ = 1\) , the authors of [1]} find \(2.496 - 1.579i\)  as the first mode, where we find \(3.223 - 1.384i\) . Our result agrees with [2]}.

For electromagnetic modes with \(\ell = 2\)  and \(r_+ = 50\)  we find \(-75.269i\)  whereas [1]} finds \(-75.139i\) . Our result agrees with [2]}.

For gravitational modes with \(\ell = 2\)  and \(r_+ = 1\) , there is some confusion over whether \(\omega = -2i\)  is a QNM. It first appears in [1]} with a question mark, it then appears in a graph in [6]}. In [2]}, the authors say they find no evidence for it. Finally it appears again in [8]} (this time without the question mark). We can verify that \(-2i\)  is indeed a QNM. In fact this mode is “algebraically special,"meaning it is purely imaginary and can be identified analytically.
It is easy to verify that \(\omega = -2i\)  satisfies \(\alpha _1^{(4)} =\alpha _1 = 0\)  and \(\beta _1^{(4)}= \beta _1 = 0\) , and thus satisfies Eq. (REF ).

For gravitational modes with \(\ell = 2\) , the authors of [2]} found \(\omega = -1.335i/r_+\)  is a good fit for the relation between \(r_+\)  and the purely imaginary \(\omega \)  for \(r_+ \ge 10\) . For \(r_+ \le 1\)  we obtain a good fit with \(\omega = 2.4416i - 4.4097i/r_+ \) . To get this fit we calculated the purely imaginary \(\omega \)  for \(r_+ = 0.3, 0.4, ..., 1.0\) .

We do not find the “highly real" QNMs that were calculated analytically in [10]}, [11]} for gravitational modes with \(\ell = 2\)  and scalar modes with \(\ell = 0\)  for large black holes. We searched in areas of the complex plane suggested by the asymptotic formula in [10]}, but found no QNMs.
However, there are issues that make searching for these modes troublesome. Our search was limited to \(\omega \)  with a relatively small real part (\(\omega _R < 65000, n\approx 500\) ) because larger values of \(\omega \)  take too long to compute. The modes only are expected to appear for large \(r_+\)  which gives rise to QNMs with large real part. Our method gets slower as the absolute value of \(\omega \)  increases because it requires a much higher depth in the continued fraction and thus higher precision in each calculation. Considering the difficulty we had finding purely imaginary modes with high absolute value, it is possible the guesses need to be very close to the actual roots to find the highly real modes.
We searched for these modes using \(r_+ = 50\) . We also searched for them by rewriting the QNM equation in the large \(r_+\)  limit (taking \(f \approx -{2M \over r}+ \frac{r^2}{R^2}\) ). In both cases we were unable to find the highly real QNMs.

